Acoustics of monodisperse open-cell foam: An experimental and numerical parametric study

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experimental and numerical parametric study

Vincent Langlois, Asmaa Kaddami, Olivier Pitois, Camille Perrot

To cite this version:

Vincent Langlois, Asmaa Kaddami, Olivier Pitois, Camille Perrot. Acoustics of monodisperse open-

cell foam: An experimental and numerical parametric study. Journal of the Acoustical Society of

America, Acoustical Society of America, 2020, 148 (3), pp.1767. �10.1121/10.0001995�. �hal-03019901�

Acoustics of monodisperse open-cell foam: An experimental and numerical parametric study

V.Langlois,^1,a)A.Kaddami,¹O.Pitois,¹and C.Perrot^2,b)

1Lab Navier, Univ Gustave Eiffel, ENPC, CNRS, F-77447 Marne-la-Vallee, France

2Univ Gustave Eiffel, Univ Paris Est Creteil, CNRS, MSME UMR 8208, F-77454 Marne-la-Vallee, France

ABSTRACT:

This article presents an experimental and numerical parametric study of the acoustical properties of monodisperse open-cell solid foam. Solid foam samples are produced with very good control of both the pore size (from 0.2 to 1.0 mm) and the solid volume fraction (from 6% to 35%). Acoustical measurements are performed by the three- microphone impedance tube method. From these measurements, the visco-thermal parameters—namely, viscous permeability, tortuosity, viscous characteristic length, thermal permeability, and thermal characteristic length—are determined for an extensive number of foam samples. By combining Surface Evolver and finite-element method cal- culations, the visco-thermal parameters of body centered cubic (bcc) foam numerical samples are also calculated on the whole range of solid volume fraction (from 0.5% to 32%), compared to measured values and to theoretical model predictions [Langlois

et al.

(2019). Phys. Rev. E

100(1), 013115]. Numerical results are then used to find approxi-

mate formulas of visco-thermal parameters. A systematic comparison between measurements and predictions of the Johnson-Champoux-Allard-Lafarge (JCAL) model using measured visco-thermal parameters as input parameters, reveals a consistent agreement between them. From this first step, a calculation of the optimal microstructures maximizing the sound absorption coefficient is performed.

V^C 2020 Acoustical Society of America.

https://doi.org/10.1121/10.0001995

(Received 27 April 2020; revised 17 August 2020; accepted 3 September 2020; published online 29 September 2020)

[Editor: Kirill V. Horoshenkov] Pages: 1767–1778

I. INTRODUCTION

Foam is a dispersion of gas in a liquid or solid matrix.

Its structure consists of membranes, ligaments (intersection of three membranes), and vertices (intersection of four liga- ments) (Fig. 1). Whereas closed membranes are necessary to ensure the mechanical stability of liquid foam (Cantat

et al.,

2013), they can be open in solid foam, allowing for the foam cells (pores) to be connected through windows.

Solid foams find applications in many fields, such as mechanical dampers, thermal, or/and acoustic insulation heat exchangers. Therefore, numerous works are still devoted to elucidate the link between their microstructure and their macroscopic properties, such as thermal, acousti- cal, mechanical, and transport properties (Despois and Mortensen, 2005; Doutres

et al., 2011;

Gibson and Ashby, 1997; Jang

et al., 2008;

Kumar and Topin, 2014). Recently, mineral solid foams, made from cement or geopolymer or gypsum, have been studied for their applications as building materials, requiring a good compromise between mechani- cal strength, and sound and/or thermal insulation (Chevillotte

et al., 2013;

Feneuil

et al., 2019;

Hung

et al.,

2014; Kaddami and Pitois, 2019; Zhang

et al., 2014). This

kind of solid foam differs from polymeric foams, which are

widely used for sound and/or thermal insulation by both their wide range of porosity (from 0.65 to 0.95) and their highly stiff porous frames.

In acoustics of porous media, semi-phenomenological models, such as the Johnson-Champoux-Allard-Lafarge (JCAL) model (Champoux and Allard, 1991; Johnson

et al.,

1987; Lafarge

et al., 1997), link the frequency-dependent

macroscopic acoustic behavior to visco-thermal parameters resulting from the microstructure. These visco-thermal parameters were introduced to properly describe the asymp- totic behaviors at high and low frequencies of the dynamic effective density for viscous parameters, and of the dynamic bulk modulus for thermal parameters. For the JCAL model, the viscous parameters are the static viscous permeability

k0

, the high frequency tortuosity

a1

and the viscous charac- teristic length

K, and the thermal parameters are the thermal

permeability

k⁰₀

and the characteristic length

K⁰

. Numerous studies have modeled these visco-thermal parameters by using upscaling methods for various microstructural config- urations. The visco-thermal parameters of idealized open-cell foams, made of 14-sided cells having spheres for vertices and circular cylinders for ligaments, were calcu- lated by finite-element method (FEM) with the aim of pre- dicting the acoustic behavior of highly porous foam (Perrot

et al., 2012). The effects on foam acoustical properties of

the membrane aperture size separating neighboring pores were studied by using models for visco-thermal parameters (Doutres

et al., 2011,

2013), or FEM calculations (Hoang

a)Electronic mail: vincent.langlois@univ-eiffel.fr, ORCID: 0000-0003- 3633-4144.

b)ORCID: 0000-0001-7796-2118.

and Perrot, 2012; Trinh

et al., 2019). However, although the

visco-thermal parameters can be characterized experimen- tally (Olny and Panneton, 2008; Panneton and Olny, 2006), many numerical studies have no systematic experimental data that can be compared with their predictions. Moreover, such theoretical studies are based on very idealized pore microstructure, which is questionable when it comes to assess the behavior of real materials. Thus, the use of more realistic microstructures should be preferred for the calcula- tion of visco-thermal parameters, and in particular in the context of researching the optimal microstructure maximiz- ing the sound absorption (Chevillotte and Perrot, 2017). To achieve this task, the software “Surface Evolver” (SE) turns out to be an excellent tool, because it has proven possible to reproduce the foam microstructure produced from gas dis- persion within a fluid (Jang

et al., 2008;

Kraynik

et al.,

2003, 2004).

The present paper deals with the acoustical properties of monodisperse open-cell foams (i.e., all pores have the same size, and foam contains no membrane). Numerical simulations combining SE and FEM are used to calculate the visco-thermal parameters of body centered cubic (bcc) foam samples (Fig. 2). In contrast to previous studies, these calculations are systemically compared to measurements performed on real foam samples.

This paper is organized as follows. In Sec. II, methods used to produce real foam and to generate numerical samples are presented, as well as the comparison of their respective microstructures. Section III is devoted to the visco-thermal parameters, where a comparison between experimental and numerical values is presented. The results are discussed and compared to dedicated theoretical models. In Sec. IV, we focus on the acoustic properties and sound absorption coeffi- cient (SAC) especially. The JCAL model predictions of

SAC

are compared to experimental measurements. Finally, calcu- lations of the optimal microstructures maximizing the sound absorption coefficient average predicted by the JCAL model are presented.

II. REAL SOLID FOAM AND NUMERICAL SAMPLES:

PRODUCTION AND MICROSTRUCTURE CHARACTERIZATION

A. Monodisperse geopolymer foam

A millifluidic setup, described in detail in Kaddami and Pitois (2019), is used to generate monodisperse foam (Fig. 3). Based on two main steps, it is able to control both the bubble size and the gas volume fraction separately. In the first step, a precursor aqueous dry foam is formed by forcing a surfactant solution flow and a nitrogen gas flow to converge in a small T-junction. The size of bubbles is fixed by adjusting the relative flow rates of solution and gas. The precursor foam is stored in a vertical glass column where a controlled foam drainage occurs. At the column exit, the gas volume fraction

/_g

is around 99%. In a second step, a meta- kaolin suspension is mixed with the precursor foam in a sec- ond T-junction. The final gas volume fraction is fixed by adjusting the relative flow rates of the metakaolin suspen- sion and precursor foam. At the exit of the second T- junction, the metakaolin fresh foam is introduced in a mold.

When filled, the mold is closed to prevent water evaporation and, the foam samples are placed at 20

C for a week. The

FIG. 2. (Color online) Bcc foam PUC as predicted by Surface Evolver for two solid volume fractions. Membranes separating the two neighboring bubbles were removed. Note that at low solid volume fraction (/_sⱗ0:11), the PUC has two kinds of windows: square- and hexagon-like windows.

And when/_sⲏ0:11, only the original hexagon-like windows remain open.

FIG. 3. (Color online) Scheme of the foaming process used to produce the geopolymer foam samples.

FIG. 1. (Color online) Close-up of foam samples revealing the open-cell microstructure and showing pores (continuous lines) and windows apertures (dotted lines). Left: Pore diameter D_p800lm;/_s0:08; right: Pore diameterDp800lm;/_s0:3.

1768 J. Acoust. Soc. Am.148(3), September 2020 Langloiset al.

solid foam sample can then easily be removed from its mold. In that stage, the films separating neighboring bubbles in foam are breaking as evidenced by a characteristic emission noise and by a binocular microscope observation (Fig. 1). To remove an amount of the initial water still present in the foam pore space after the metakaolin geopolymer setting, foam sam- ples must be dried. After drying, acoustical measurements are performed, and the microstructure of the geopolymer foam samples is characterized by binocular microscope observations and for a few samples by x-ray microtomography.

Ensuring foam stability is a key point during each step of the whole process. Concerning the precursor foam pro- duction, its duration depends on the size of the bubbles, the smaller the bubbles are, the longer the time of production is.

To prevent the ripening in the case of the smallest sizes (0.2 and 0.3 mm), nitrogen used as bubbling gas in the precursor foam production was saturated in perfluorohexane vapor (Gandolfo and Rosano, 1997). To ensure the stability of the fresh foam before the geopolymer setting, particular atten- tion was paid to the choice of geopolymer formulation, as shown in detail in Kaddami and Pitois (2019). However, despite our special care, a small and unexpected drainage was observed for high solid volume fraction (/

_sⲏ

0:3) lead- ing to a low gradient of solid volume fraction.

Open-cell foams with monodisperse pore diameter

Dp

ranging from 0.2 to 1.0 mm (0.2, 0.3, 0.6, 0.8, and 1.0 mm) and solid volume fraction ranging from 0.1 to 0.38 were thus produced. X-ray microtomography image analysis shows that the relative standard deviations for the pore diameter are around 5% for the smallest diameters and up to 15% for the largest ones.

B. PUC numerical samples

As shown in Fig. 2, a periodic unit cell (PUC) is used to represent the pore structure in foam samples. The cell is based on the bcc paving. The unit cube of size

Dt

includes two pores (Fig. 2): one located in the center of the cube, and the other split into eight parts at the vertices of the cube. The SEF-FIT software, which runs SE as computational engine (Brakke, 1992), was used to compute the shapes of bcc foam samples by minimizing their surface tension energy. The fine and deep convergence algorithms within SEF-FIT were used to approach the shape of the equilibrium foam surfaces. Before the last and deepest convergence procedure, an edge length threshold equal to 0:02D

t

was used (i.e., all the edge lengths are smaller than this length threshold) leading to a mean edge length of about

ð0:0126

0:002ÞD

t

. After the SE calculations, membranes separating the neighboring pores were removed to obtain a fully open-cell foam. Figure 2 shows two examples of solid pore shapes obtained at

/_s¼

0:04 and 0.12. For low solid volume fraction (/

_sⱗ

0:11Þ, the structure corresponds to the so-called Kelvin structure (i.e., each bubble is linked to 14 bubbles through two kinds of windows: square-like win- dows and hexagon-like windows). The aperture size is defined by an equivalent diameter of window aperture,

do¼ ðð4=pÞSoÞ¹⁼²

, where

So

is the surface area of the

membrane predicted by SEF-FIT calculations. The aperture sizes of both types of window,

do;sq

for square-like windows and

do;hex

for hexagon-like windows, are given in Fig. 4. As the solid volume fraction increases, the aperture sizes of the windows decrease, leading to a closure of square-like win- dows near to

/_s

0:11. The full closure of the original hexagon-like windows, leading to closed-cell foam (when

/¼/^?_s

, the critical solid volume fraction), is reached when the pore becomes spherical for

/_s ¼/^?_s;BCC¼

1

ð ffiffiffi

p

3

=8Þp

0:32 (for bcc packing of spheres). These values (/

_s

0:11 and

/_s

0:32) are in agreement with previous results found by Murtagh

et al.

(2015).

We define a bubble size (or pore size)

Dp

as the diame- ter of a sphere having the same volume as the bubble:

Dp

¼ ðð3=pÞð1/_sÞÞ¹⁼³Dt

.

C. Microstructure of real foam samples compared to numerical bcc foam

In addition to solid volume fraction and pore size, the mean size

hdoi

of apertures between pores and the mean number

Nv

of open windows per pore are important micro- structural parameters for macroscopic properties such as permeability and tortuosity (Despois and Mortensen, 2005;

Johnson

et al., 1987;

Kirkpatrick, 1973; Langlois

et al.,

2018, 2019). For some real foam samples, binocular micro- scope observations were used to determine the distribution of the aperture sizes. For each sample, around fifty apertures were observed in order to calculate the mean aperture

hdoi_So

. Figure 4 shows the evolution of the weighted mean aperture as a function of the solid volume fraction. As shown for the numerical geometries, the aperture size decreases as the solid volume fraction increases. However, for real samples, open windows were observed for the solid volume fraction higher than

/^?_s;BCC

. Due to the small

FIG. 4. (Color online) Diameter of aperturesdodivided by the pore diame- terDp as a function of the solid volume fraction/_s. The dashed line is drawn by using Eq.(1). The error bars correspond to the standard deviation.

Note that for bcc numerical samples, the aperture size of hexagon- and square-like windows and the mean aperture sizehdoiare distinguished. The aperture diameter is defined as the diameter of a disk having the same area as the membrane area So closing the windows after SE calculations:

d_o;i¼ ðð4=pÞSo;iÞ¹⁼², where indexiis associated to hexagon- or square-like windows.

drainage detected in real foam samples for high solid vol- ume fraction, an accurate measurement of the critical solid volume fraction

/^?_s

is not possible. However, the results of our experiments suggest that

/^?_s;exp

0:38. This value is close to the common value found in the literature

/^?_s

0:36 for monodisperse and disordered foam (Drenckhan and Hutzler, 2015). Based on our data presented in Fig. 4, we propose the following expression to describe the measured aperture size over the full range of solid volume fraction:

hdoi_So ¼Dp /^?_s/_s0:5

1

/^?_s/_s 0:5

n

3:75/s/^?_s/_s

(1)

For few samples, an analysis of x-ray microtomography images using ImageJ software, described in detail in Pitois

et al.

(2020), allowed us to identify the pores (their centers and their sizes) and to locate the apertures between them. To check the location of the apertures, a calculation is per- formed by testing for each pair of adjacent pores if an aper- ture was detected in the vicinity of its expected position (defined by the barycenter of the pore centers weighted by the pore sizes). If such an aperture is found in the list of

“aperture” objects, this aperture is considered as being iden- tified. At the end of the identification process, the mean number

Nv

of open windows per pore is calculated. Figure 5 shows the mean number of open windows per pore found for four samples. Experimental values are close to the values determined from numerical simulations, showing a decrease of

Nv

as the solid volume fraction increases. Our experimen- tal data for

Nv

are few but they can be combined with expected bounds [i.e.,

Nv

14 for

/_s

0 (Kelvin cells) and

Nv

6 for

/_s/^?_s;exp

]. Figure 5 shows that

Nvð/_sÞ

can be described by the following linear relation:

Nv¼

8 1

/_s /^?_s

þ

6 (2)

Finally, the analysis of the pore positions by computing the pair distribution function did not reveal any order in our

samples, contrary to the bcc numerical samples. Therefore, except for this major difference between real foam samples and bcc numerical samples, the microstructure of SE bcc numerical samples appears quite similar to that of the real foam samples in terms of

Nv

and

do

behavior (except for

/_s

0:32).

III. VISCO-THERMAL PARAMETERS A. FEM calculations on PUC samples

In this section, we briefly introduce the boundary value problems used for computing the visco-thermal parameters: (1) viscous parameters: the static viscous per- meability

k0

, the high frequency tortuosity

a1

, and the vis- cous characteristic length

K, (2) thermal parameters:

the static thermal permeability

k⁰₀

and the thermal charac- teristic length

K⁰

. The solution of the boundary value prob- lem is obtained through the FEM using COMSOL Multiphysics software.

In what follows, the distinction between the viscous flow and the inertial flow is whether

xxc

or

xxc

, respectively, with

xc¼l/_g=ðq₀a1k0Þ, where l

is the dynamic viscosity of the fluid,

q₀

is the fluid density at rest, and

/_g

is the gas volume fraction (Pride

et al., 1993).

1. Viscous flow

The low Reynolds number flow of an incompressible Newtonian fluid is governed by the usual Stokes equations in the fluid phase:

l䉭v rp¼

0 with

r v¼

0 in

Xf; v¼

0 on

@Xp;

vn¼

0 on

@Xf;lateral;

pbottomptop¼DP

on

@Xf;extremety;

v

is

Dt

periodic on

Xf;

(3)

where the symbols

v

and

p

stand for the microscopic veloc- ity and pressure of the fluid, respectively;

pbottom

and

ptop

are the pressures on the top and bottom faces of the PUC;

DP

is the macroscopic pressure difference acting as a forcing term; and the fluid boundaries

@Xp

(pore surface),

@Xf;lateral

, and

@Xf;extremety

are defined in Fig. 6.

The macroscopic pressure gradient

km

is defined as

km¼DP=Dt

.

The static viscous permeability

k0

is calculated from the microscopic velocity averaged over the pore fluid volume,

hvi_Xf

,

k0¼lhvi_Xf e km

;

(4)

where

e

is the unit vector pointing in the direction of the macroscopic pressure gradient, and

hi_Xf

denotes averaging over the pore fluid volume

hi_Xf ¼ ð1=VfÞÐ

XfdVf

.

FIG. 5. (Color online) Mean number of apertures per poreNvas a function of the solid volume fraction/_s. The dashed line is drawn by using Eq.(2).

Note that for bcc foam, the discontinuity ofNvat/_s0:11 is due to the closure of the six original square-like windows.

1770 J. Acoust. Soc. Am.148(3), September 2020 Langloiset al.

2. Inertial flow

At the high frequency range, the viscous boundary layer becomes negligible, and the fluid tends to behave as a per- fect fluid (or inviscid fluid) and to flow through the pore space as electric charges would migrate through the pore space if it was filled by a conducting fluid having a constant conductivity. Consequently, the perfect incompressible fluid formally behaves according to the electrical conduction problem (Brown, 1980; Johnson

et al., 1987):

䉭u¼

0 with

E¼ ru

in

Xf; En¼

0 on

@Xp

and

@Xf;lateral;

u_bottom¼ u_top ¼ DV=2 on @Xf;extremity;

u

is

Dt

antiperiodic on

Xf;

(5) where

E

and

u

are the local electric field and the electric potential, respectively, and

n

is the unit vector normal to

@Xp

and

@Xf;lateral

.

The high frequency tortuosity

a1

and the viscous char- acteristic length

K

are calculated as follows (Cortis

et al.,

2003; Johnson

et al., 1987):

a1¼ hEEi_Xf

hEi_Xf hEi_Xf ¼

1

hEi_Xf e;

(6)

K¼

2

ð

Xf

EEdVf

ð

@Xp

EEdSp

;

(7)

where

e

corresponds to an unit global (i.e., external, locally constant) electric field.

Because of the sharp edge existing at the level of the throat between two neighboring pores [Figs. 7(a) and 7(b)], the electric field has a singularity along the edge. To reduce the lack of accuracy in the numerical approximation of

E,

special care was taken to refine the mesh in the region of the sharp edge (Cortis

et al., 2003;

Firdaouss

et al., 1998). The

maximum element size

h

along the throat edge was imposed [Fig. 7(c)]. Therefore, this size corresponds to the spatial resolution of our FEM calculations (i.e., our FEM results are similar to those with a rounded edge of radius equal to

h).

As we are interested in geopolymer foam made from micro- metric metakaolin particles and having millimetric pores, we choose

h¼Dt=1000.

3. Thermal parameters

The static thermal permeability is given by Henry and Allard (1997) and Lafarge

et al.

(1997):

k⁰₀¼ ð1/_sÞhhi_Xf;

(8) where

h

corresponds to the scaled excess temperature field and solves the following equations:

䉭h¼ 1 inXf; h¼

0 on

@Xp;

rhn¼

0 on

@Xf;lateral

and

@Xf;extremity

(9)

The thermal characteristic length is a geometric param- eter, depending on the shape of the pore space:

K⁰¼

2

ð

Xf

dVf

ð

@Xp

dSp

(10)

It is to be noted that when the solid volume fraction is high enough, the pores are almost spherical. In such a case, the thermal characteristic length and the static thermal per- meability can be calculated by the following formulae, lim

_/s!/^?

sK⁰=Dp¼¹₃

and lim

_/s!/^?

sk⁰₀=ð1/_sÞD²_p¼₆₀¹

(details are given in Appendix C).

FIG. 6. (Color online) Geometry of the pore volume and definitions of the fluid boundaries (/_s¼0:12).

FIG. 7. (Color online) (a) Geometry of the solid frame. The edges of throats are highlighted by a red thick line. (b) Cross-section of a ligament. (c) Pore volume mesh of 1/16 Kelvin cell for FEM calculation of the inertial flow (/_s¼0:08).

B. Acoustical characterization of real foam samples

We determined the macroscopic parameters of real foam samples by acoustic measurements performed in a three-microphone impedance tube (length: 1 m, diameter:

40 mm) (Doutres

et al., 2010). The tested frequency range

lies between 4 and 4500 Hz, with a step size of 4 Hz. The three-microphone impedance tube method consists in mea- suring the pressure transfer functions between the micro- phones and leads, after calculations, to the dynamic density

qeq

and the dynamic bulk modulus

Keq

of the equivalent fluid medium. The sound absorbing coefficient at normal incidence

SACNI

directly derives from

q_eq

and

Keq

. Based on the measured frequency dependent response functions

qeq

and

Keq

, Panneton and Olny (2006; Olny and Panneton, 2008) proposed an inverse characterization method to esti- mate the visco-thermal parameters of porous materials. This characterization method requires the porosity value and the static viscous permeability

k0

as input parameters. However, by using the approximate but robust JCAL semi- phenomenological model (Champoux and Allard, 1991;

Johnson

et al., 1987;

Lafarge

et al., 1997), it is possible to

determine the static viscous permeability

k0

. Indeed, the JCAL model predicts that the imaginary part of

qeq

is given by

Im

ðq_eqÞ ¼ l k0x

1 2

þ

1

2 1þ

M

2x

_lx 2!0:5

2 4

3 5

0:5

(11)

In the aim of finding the static viscous permeability

k0

,

x⁰⁰

(¼

M=2x_l

) and

k0

are chosen to fit the model to data.

Details of the JCAL model are given in Appendix A.

C. Pore-network models for permeability and tortuosity

Recent studies proposed pore-network models for per- meability and electrical conductivity of solid foam (Langlois

et al., 2018,

2019). For permeability, the model considers that the fluid flow through solid foam is governed by the pressure drops that occur when the fluid flows through the windows. A local fluid flow conductance is then used to describe these pressure drops (Sampson, 1891). The model describes the foam pore-space as a network of fluid flow conductances. By solving a problem similar to the calculation of the equivalent electrical resistance of an elec- trical resistance network, it is possible to solve the pore- network of fluid flow conductance and to deduce the foam permeability. This pore-network model has been success- fully validated for highly porous foam having thin open membranes by comparing its predictions to PUC FEM cal- culations (Langlois

et al., 2018). For tortuosity, the problem

is more difficult to solve because the electrical conductance of solid foam does not only rely on a local mechanism, such as the local pressure drops for viscous fluid flow (Langlois

et al., 2018). In this case, the whole pore volume also con-

tributes to the overall electrical conduction. However, when

the apertures are small compared to the pore size, the access resistances acting at the scale of the windows’ apertures are predominant compared to the bulk electrical resistance of the pore. As a consequence, the problem for electrical con- ductivity is very similar to the ones for permeability.

Hereafter, we recall the expressions for permeability and tortuosity (where the bulk electrical resistance of the pore is neglected) regarding foam having a bcc structure (Langlois

et al., 2018,

2019):

k0

D²_p¼

1 24

Dp

Dt

nsq

do;sq

Dp

3

þ

1 2

nhex

do;hex

Dp

3

" #

;

(12)

1

a1

1 1

/_s

Dp

Dt

nsq

do;sq

Dp

þ

1 2

nhex

do;hex

Dp

;

(13)

where

nsq¼

2 (while

/_sⱗ

0:11, 0 afterward) and

nhex¼

4 correspond to the numbers of square- and hexagon-like win- dows through which the flow effectively occurs in bcc foam.

Note that the dependence of the permeability on the aperture size (/

d³_o

involved by Sampson’s law) and the one of the tortuosity to the aperture size [/

do

involved by the access resistance given by Hall’s law (Hall, 1975)] come from the local conductances (or resistances) acting at the scale of the windows’ apertures (Langlois

et al., 2019).

When the aperture size and the pore size are in the same order of magnitude, both volume and access resistances must be taken into account to calculate the tortuosity. Here, we give the expression of tortuosity adapted for the case of bcc open-cell foam (Langlois

et al., 2019):

1

/_s a1

¼re

rf

¼

2 2g

1g2þge;hðg1þg2Þ g1þg2þ

2g

e;h

;

(14)

where

g1¼ ðg¹_i;shþ

2g

¹_e;sÞ¹

and

g2¼

2ðg

i;shþgi;hhÞ

with

gi;sh

¼0:16; gi;hh¼

0:30;

ge;i¼ðdoi=DtÞð1þC1ðb_iðdoi=DtÞÞ þC2ðb_iðdoi=DtÞÞ³Þ¹

where

b_i¼ ffiffiffiffiffiffi

p

2p

for square windows (i

⁰s⁰

) and

ð4p=3^1:5Þ^0:5

for hexagonal windows (i

⁰h⁰

), and

C1¼ 1:265 andC2¼

0:166.

In Eq. (14),

ge;s

and

ge;h

are the inverses of access resis- tances associated to square and hexagonal windows, respec- tively, and

gi;hh

and

gi;sh

are the electrical conductances associated to the bulk pore contribution. In the limit case

doi!

0, Eqs. (13) and (14) are identical.

D. Results and discussion

The visco-thermal parameters are gathered in Figs. 8, 9, and 10, as calculated by FEM on bcc PUC and characterized for real foam samples.

Hereafter, we consider the consistency of our numerical and experimental data in the light of theoretical models.

1. Tortuosity and viscous permeability

Viscous permeability

k0

and tortuosity

a1

are two parameters governing the asymptotic behavior of the appar- ent dynamic density of the fluid at low frequency for

k0

and at high frequency for

a1

. As mentioned in Sec. III A, both

1772 J. Acoust. Soc. Am.148(3), September 2020 Langloiset al.

parameters involve a dedicated transport phenomenon: the flow of a viscous fluid at low frequency and the displace- ment of electric charges at high frequency, respectively.

Therefore, the tortuosity at high frequency is linked to the foam electrical conductivity

re

when the pore space is filled by a fluid having an electric conductivity

rf

:

a1¼ ð1/_sÞrf=re

[see, for instance, Eq. 2.8 in (Johnson

et al.,

1987)]. Thus, as the solid volume fraction increases, the ability of fluid or electrical charges to pass through the foam samples decreases due to the progressive closure of the win- dows (r

_e

decreases strongly when

/_s

increases), and until vanishing when

/_s!/^?_s

. Consequently, as shown in Figs.

8 and 9, tortuosity is expected to increase when the solid volume fraction increases, while the viscous permeability decreases.

Figure 9(a) shows that the predictions of the pore- network model for permeability is in perfect agreement with the FEM calculations except for very low solid volume frac- tion where a discrepancy is observed. Concerning tortuosity [Fig. 8(a)], the pore-network model predicts well the ten- dency calculated by FEM but leads to slightly smaller values

even at high solid volume fraction. This gap is not surprising at low solid volume fraction, because the resistance term taking into account the bulk electrical resistance should be added. Taking into account this contribution [Eq. (14)]

allows us to improve the agreement [see Fig. 8(a)] while much more complexity is introduced in the model.

However, a small discrepancy between FEM calculations and theoretical predictions remains at high solid volume fraction. It could be due to the effect of the sharp edge of throats, which alters the local electrical field singularity, and consequently, access resistances.

2. Other parameters

To our knowledge, no theoretical models have been pro- posed yet for the viscous characteristic length and thermal parameters of solid foam. The definitions of thermal and vis- cous characteristic lengths involve a ratio of two integrals over the pore volume at the numerator and over the pore sur- face at the denominator. In solid foam samples, the pore

FIG. 8. (Color online) Tortuositya1as a function of the solid volume frac- tion/_s: (a) FEM on PUC numerical samples (triangle). (b) Real foam sam- ples for various pore sizes (markers, see legend). The red dashed lines in (a) and (b) are drawn by using Eq.(16)(with /^?_s0:32 for bcc numerical foam samples, and/^?_s0:38 for real foam samples). In (a), dash-dotted and full lines correspond to the pore-network model predictions with the bulk pore contribution for the blue dash-dotted line [Eq.(14)] and without for the black full line [Eq.(13)].

FIG. 9. (Color online) Viscous permeabilityk0and thermal permeabilityk₀⁰ as a function of the solid volume fraction/_s: (a) FEM on PUC numerical samples (triangle), (b) Real foam samples for various pore sizes (markers, see legend). The red dashed lines in (a) and (b) are drawn by using Eq.(16) (with/^?_s0:32 for bcc numerical foam samples, and/^?_s0:38 for real foam samples). In (a), the black full line corresponds to the pore-network model predictions: Eq. (12). Note that for spherical pores, k⁰₀=D²_p

¼ ð1/_sÞ=60 (green dashed-dotted line).

volume does not evolve significantly as the solid volume frac- tion increases compared to the pore surface: on the one hand,

Ð

@XfdSf !

0, when

/_s!

0; whereas on the other hand,

Ð

XfdVf !Cst, when/_s!

0 (see Figs. 1 and 2). Therefore, as shown in Fig. 10(a), each characteristic length diverges when

/_s

tends to zero and decreases as

/_s

increases. This result is in qualitative accordance with the expected behavior for fibrous materials. For such materials, Allard and Atalla (2009) shows that

K¼K⁰=2¼

1=2pR

0L

where

L

is the total length of fibers per unit volume of fibrous material for cylin- drical fibers of radius

R0¼ ð/s=pLÞ^0:5

in the dilute limit (no interaction between fibers). Therefore, for fibrous materials with a fixed

L, both characteristic lengths diverge when /_s!

0. In the limit of low

/_s

, the foam microstructure is not far from the microstructure of fibrous materials as ligaments are very thin and elongated. This microstructural analogy between fibrous material and foams when

/_s!

0 is consis- tent with the ratio

K⁰=K

being close to two for both the simu- lated and experimental foam samples. However, the relations

K¼K⁰=2¼

1=2pR

0L, which is correct for fibrous media (in

the dilute regime), are not directly applicable for foams, even

when

/_s!

0, because of the non-circular cross-section shape of the ligaments. Moreover, for high solid volume fraction, the ratio

K⁰=K

diverges because

K⁰!¹₃Dp

[“sphere limit,”

Fig. 10(a)] and

K!

0. Therefore, a foam specific model for each characteristic length remains to be found. As the thermal characteristic length

K⁰

is a pure geometrical parameter depending on the real shape of foam pore, no simple geomet- ric model can be used to evaluate this parameter. Numerical calculations using SE appear to be appropriate to find a realis- tic relationship between

K⁰

and

/_s

. A similar approach was used by Pitois

et al.

(2009) to calculate the volume specific surface area

SV

as a function of the liquid volume fraction

/_liq

in the context of the study of liquid foam drainage study (note that

SV

2=K

⁰

). Concerning the viscous characteristic length

K, its definition involves a pore-volume-to-surface ratio

weighted by the square of the electrical field [cf. Eq. (7)]. As the electrical field is very intense along the sharp edge of throats (cf. Sec. III A), this area of the pore volume is favored by the weighted average procedure. Therefore, for high solid volume fraction, the viscous characteristic length is compara- ble to the size of the window aperture [Fig. 10(b)]. However, this is no longer applicable when the solid volume fraction is low due to the low surface area of pores. Clearly, the viscous characteristic length seems to be a balance between the size of the window aperture and the ligament thickness (or the pore surface area). These considerations lead us to propose the fol- lowing expression for

K, which was found to give a very good

agreement with numerical values [see Fig. 10(a)]:

K Dp

D²_thdoi_So Sp=pore

;

(15)

where

Sp=pore

corresponds to the pore surface area of a single pore (i.e., the pore surface area contained in the PUC divided by two), and

hdoi_So

is the area-weighted mean diam- eter of apertures.

Concerning the thermal permeability, Fig. 9 shows that

k₀⁰=D²_p!₆₀¹ð1/_sÞ

when

/_s!/^?_s

(as expected for spheri- cal pores). For low solid volume fraction, the thermal per- meability predicted by our FEM results can be compared to the ones obtained for fibrous materials (cf. Appendix B).

This model predicts that

k⁰₀/ lnð/sÞ

when

/_s!

0.

Therefore, the thermal permeability diverges when

/_s!

0.

For real foam samples, results are significantly dispersed around the values calculated by FEM.

3. Approximate formulas

For each visco-thermal parameter, a systematic compari- son between FEM calculations on bcc numerical foam samples and real monodisperse foam characterization reveals a good agreement between them except for high solid volume fraction due to the discrepancy in the critical solid volume fraction

/^?_s

. Approximate formulas for visco-thermal parameters are derived from our numerical results (Figs. 8, 9, and 10):

FIG. 10. (Color online) Viscous characteristic lengthKand thermal charac- teristic lengthK⁰as a function of the solid volume fraction/_s: (a) FEM on PUC numerical samples (triangle). (b) Real foam samples for various pore sizes (markers, see legend). The dashed lines in (a) and (b) are drawn by using Eq.(16)(with/^?_s0:32 for bcc numerical foam samples, and/^?_s 0:38 for real foam samples). In (a), full line forKis drawn by using Eq.(15). Note that for spherical pores,K⁰=D_p¼¹₃(dashed-dotted line).

1774 J. Acoust. Soc. Am.148(3), September 2020 Langloiset al.

k0

D²_p

exp

ð68:617x⁵þ148:54x⁴124:36x³ þ49:897x²13:701x3:532Þ;

k⁰₀ D²_p

1

60

0:68lnð Þþ0:48x /^?_sþ0:7x0:18x²

;

K Dp

0:087x

^1:16ð

1x

Þ^0:37þ0:065x^0:37ð

1x

Þ^1:16

x¹;

K⁰ Dp

1

3

0:85ð Þx^0:94þ1:85ð Þx^0:5

1

;

a1ð

1/

sÞ

0:47 1x

ð Þ^2:47þ0:53 1xð Þ^0:51

;

(16) where

x¼/_s=/^?_s

.

Figures 8, 9, and 10 show that these formulas can be used to estimate the values of visco-thermal parameters of monodisperse foam samples on the whole range of

/_s

by choosing the appropriate critical solid volume fraction

/^?_s

(/

^?_s

0:32 for bcc numerical foam samples, and

/^?_s

0:38 for real foam samples). These formulas are only valid for the case of monodisperse open-cell foams, because their dis- persions in pore sizes, aperture size, and number of aper- tures per pore are low. Note that the formula for

K

is established for a maximum element size

h

(spatial resolution of the throat edge) equal to 0:001D

t

. For other spatial resolu- tions, our FEM calculations performed with various values of

h=Dt

and

/_s¼

0:2 show that a correction must be applied to better predict

K: Kðx;h=DtÞ=DpKðx;

0:001Þ=

ð1

0:103 log

ðh=DtÞÞ. These formulas could be useful in

the context of the characterization of visco-thermal parame- ters of foam samples, or in the context of material optimiza- tion, as illustrated in Sec. IV.

IV. NORMAL INCIDENCE SOUND ABSORPTION

The present section focuses on the sound absorption of foam materials by presenting our measurements performed on real foam samples and by comparing them to the predic- tions of the JCAL model. The JCAL model is one of the most popular semi-phenomenological models of the acoustic properties of rigid porous materials. The constitutive equa- tions of this model are recalled in Appendix A. The JCAL model agrees closely with the experimental data of dynamic bulk modulus

Keq

and dynamic density

q_eq

of the equivalent fluid medium (see supplementary material).

¹

At the end of this section, by using the previously established approximate formulas to calculate the input parameters of the JCAL model, an optimization calculation of normal incident sound absorption coefficient

SACNI

will be presented.

Figure 11 shows that foam samples of thickness between 20 and 30 mm with low solid volume fraction and a pore size around 0.3 mm have high sound absorption coeffi- cients on a large frequency range. However, as sound absorption depends on both the pore size and the solid vol- ume fraction, other configurations on

/_s

and

Dp

leading to high sound absorption could be possible (at a given thick- ness, with a rigid backing). Predicting the effect of each

parameter on SAC is not straightforward. For example, Fig. 11(a) shows that the maximal value of the sound absorption coefficient does not depend monotically on the pore size at constant solid volume fracion

/_s

(i.e., at con- stant ratio

hdoi_So=Dp

). Indeed, the sound absorption coeffi- cient measures the amount of energy, which is backward in the incident direction when the material is backed by an impervious rigid wall. Therefore, this coefficient depends on the ability of the incident sound to pass through the first air/

foam interface and on the ability of the porous material to dissipate the sound energy. A highly porous material favors the first mechanism but not the second. Thus, a good balance between these two mechanisms has to be found to maximize the sound absorption. Well-validated theoretical models are therefore expected to be helpful in finding the optimal con- figuration for

/_s

and

Dp

. Figure 11 shows that the predic- tions of the JCAL model may be quite relevant to pursuing this objective. In order to provide a global indicator of the

SACNI

over a large frequency range, we use an average of

SACNI

by one-third octave bands

fi

from 125 to 4000 Hz:

FIG. 11. (Color online) Sound absorption coefficient at normal incidence SACNIas a function of frequency for real foam samples: (a) various pore sizesDpwith/_s0:1 (sample thickness 19 mm), (b) various solid volume fraction withDp0:3 mm (sample thickness 28 mm). Full lines correspond to the JCAL predictions plotted by using the characterized parameters. In (a), the measurement forD_p¼0:3 mm is omitted because the sample thick- ness was 28 mm.

SAA¼

1 16

X

f_i

SCANI;1=3ðfiÞ

(17)

Note that the standard ASTM C423 (ASTM C423–17, 2017) uses a smaller frequency range (200–2500 Hz).

Figure 12 compares the mean sound absorption coeffi- cient

SAA

measured on real foam samples to the ones calcu- lated with the JCAL model. It appears that the JCAL model

predicts SAC values slightly lower than those measured.

This small discrepancy could be attributed to a systematic error of the JCAL model to predict the intermediate fre- quency regime (Cortis

et al., 2003;

Pride

et al., 1993). By

using the JCAL model and the approximate formulas pro- vided in Sec. III [Eq. (16)] for visco-thermal parameters with

/^?¼

0:38, we have calculated

SAA

for various porosi- ties (/

_g¼

1

/_s

) and pore sizes, and solved the optimiza- tion problem to maximize

SAA

for two sample thicknesses.

Results are gathered in Fig. 13. As observed with measure- ments on real foam samples, the optimal configurations of

ð/_s;DpÞ

are located in a range of high porosity (/

_s<

0:1) and small pore sizes (D

p<

0:3 mm). These results are in qualitative agreement with previous results obtained by Chevillotte and Perrot (2017) with FEM calculations per- formed on idealized bcc interpenetrating spheres.

V. CONCLUSION

This paper presents an experimental and numerical parametric study of the acoustical properties of monodis- perse foam. Our numerical model combines the semi- phenomenological JCAL model and FEM calculations of visco-thermal parameters performed on bcc foam structure, calculated with SE (surface area minimization software).

A summary of the main results is provided as follows:

•

Visco-thermal parameters calculated by FEM on SE bcc foam PUC are similar to the ones measured on real mono- disperse foam samples, except for high solid volume frac- tion (/

_s>

0:25).

•

The static viscous permeability

k0

and the tortuosity

a1

calculated by our numerical simulations compare fairly well with the previously published predictions of effective medium models (Langlois

et al., 2018,

2019).

•

Approximate formulas based on our numerical results are proposed and can be used to estimate the visco-thermal parameters of real monodisperse foam samples.

•

As observed on real foam samples, the optimal configura- tions of

ð/s;DpÞ

for normal incidence sound absorption, predicted by the combination of the JCAL model and our approximate formulas, correspond to a range of high porosity (/

_s<

0:1) and small pore sizes (D

p <

0:3 mm) for a moderate sample thickness (20 25 mm).

APPENDIX A: JCAL MODEL

In this appendix, we recall the expressions of the dynamic density

q_eq

and bulk modulus

Keq

of the equivalent fluid medium as provided by the JCAL model (Champoux and Allard, 1991; Johnson

et al., 1987;

Lafarge

et al., 1997)

and expressed as in Olny and Panneton (2008; Panneton and Olny, 2006):

q_eq¼ q₀a₁ /_g þ l

k0xGI

j l

k0xGR;

(A1)

FIG. 12. (Color online) Comparison between the sound absorption averages measured on real foam samples, SAAexp, and the ones predicted by the JCAL model,SAAmodel. Filled dots correspond to 19-mm thick samples, and empty dots to 28-mm thick samples.

FIG. 13. (Color online) 2D maps of mean sound absorption coefficientSAA as a function of pore sizeDpand porosity/_g¼1/_s: (a) Sample thick- ness, 25 mm; (b) Sample thickness, 20 mm. The black thick line shows the morphological configurationsðDp;/_gÞmaximizing the mean sound absorp- tion coefficientSAA. The arrow indicates the best morphological configura- tion. Gray lines correspond toSAAcontour lines. The difference of sound absorption ratings between adjacent contour lines is 0.025. Numbers indi- cate to the values ofSAAof the thickest contour lines. In (a), the dashed line corresponds to the results obtained byChevillotte and Perrot (2017) with bcc structure of interpenetrating spheres.

1776 J. Acoust. Soc. Am.148(3), September 2020 Langloiset al.

cP0

Keq

¼cðc

1

Þ

1

jxt

x

1

þjM⁰

2

x xt

1=2!1

;

(A2) where

GR ¼ ½¹₂þ¹₂ð1þ ð^M_2x^x_lÞ²Þ¹⁼²¹⁼²

and

GI ¼ ðM=4GRÞ ðx=xlÞ,

with

M¼

8a

1k0=/_gK²

,

xl¼l/_g=k0a1q₀

,

M⁰¼

8k

⁰₀=/_gK⁰²

, and

xt¼l/_g=k⁰₀Prq₀

.

In these equations,

P0

and

q₀

are the pressure and mass density, respectively, of the saturating fluid at rest,

l

is its dynamic viscosity,

NPr

is the Prandtl number (0:71 for air),

c¼Cp=Cv

is the ratio of the heat capacities at constant pressure and volume, and

j

the imaginary unit.

The wave number

kcðxÞ

and the characteristic imped- ance

ZcðxÞ

of the equivalent-fluid are given by

kc¼x q _eq=Keq1=2

and

Zc¼q_eqKeq1=2

(A3) The sound absorbing coefficient at normal incidence

SACNI

of a layer of equivalent-fluid backed by an impervi- ous rigid wall is related to the specific surface impedance

Zs

of the sample of thickness

Hsp

:

SACNI ¼

1

Zsðx;HspÞ Z0

Zsðx;HspÞ þZ0

2

;

(A4)

where

Z0¼ ðcq0P0Þ¹⁼²

is the characteristic impedance of saturating fluid and

Zsðx;HspÞ ¼ jðZcðxÞ=/_gÞ

cotðk

cðxÞHspÞ.

APPENDIX B: THERMAL PERMEABILITY OF FIBROUS MATERIALS

In this appendix, we present the calculation of the ther- mal permeability for the case of a fibrous material made of

long and straight cylindrical fibers. To simplify the resolu- tion of the problem, the pore volume surrounding the fiber is supposed cylindrical with a “zero flux” boundary condition (Fig. 14). The solid volume fraction is given by

/_s¼R²₀= R²₁

, and the radius of the pore cylinder

R1

is linked to the total length of the fiber per unit volume of fibrous material,

L:R²₁¼

1=pL.

The solution of Eq. (10) can be found easily:

hðrÞ

¼R²₁=2

ln

ðr=R0Þ þ¹₄ðR²₀r²Þ. By averaging this solution

over the pore volume, we find the thermal permeability

k⁰₀

:

k⁰₀¼

1

8pL

h2 lnð Þ /_s

3

þ

4/

_sþ/²_si

(B1)

In the limit of low solid volume fraction, the structure of solid foam samples is quite similar to fibrous material.

If we consider a BCC foam sample made of 24 ligaments per PUC, having a length equal to

Dt=2³⁼²

, we can calculate the total length of the fiber per unit volume:

L ¼

12=2

¹⁼²D²_t

. The thermal permeability is then given by

k⁰₀

2

¹⁼²D²_t

96p

h2 lnð Þ /_s

3

þ

4/

_sþ/²_si

(B2) The approximate formula for

k⁰₀

given in Eq. (16) is based on Eq. (B2). However, coefficients inside the brackets are adjusted to have a better fit data.

APPENDIX C: THERMAL PERMEABILITY OF SPHERICAL PORES

In this appendix, we present the calculation of the thermal permeability for the limit case of a spherical pore. We con- sider that the solution of Eq. (10) has a spherical symmetry

hðrÞ, wherer

is the distance to the pore center. With straight- forward steps, we solve Eq. (10):

hðrÞ ¼ ðR²_pr²Þ=6. By

averaging this solution over the pore volume, we find the ther- mal permeability

k⁰₀=D²_p¼ ð1/_sÞ=60.

1See supplementary material at https://doi.org/10.1121/10.0001995 for comparison between measurements and JCAL model predictions of dynamic densityqeq and bulk modulus Keq.

Allard, J. F., and Atalla, N. (2009).Propagation of Sound in Porous Media (John Wiley and Sons, West Sussex, UK).

ASTM C423-17. (2017). “Standard Test Method for Sound Absorption and Sound Absorption Coefficients by the Reverberation Room Method”

(ASTM International, West Conshohocken, PA).

Brakke, K. A. (1992). “The surface evolver,”Exp. Math.1(2), 141–165.

Brown, R. J. S. (1980). “Connection between formation factor for electrical resistivity and fluid-solid coupling factor in Biot’s equations for acoustic waves in fluid-filled porous media,”Geophysics45(8), 1269–1275.

Cantat, I., Cohen-Addad, S., Elias, F., Graner, F., H€ohler, R., Pitois, O., Rouyer, F., and Saint-Jalmes, A. (2013).Foams: Structure and Dynamics (Oxford University Press, Oxford).

Champoux, Y., and Allard, J. F. (1991). “Dynamic tortuosity and bulk mod- ulus in air-saturated porous media,”J. Appl. Phys.70(4), 1975–1979.

Chevillotte, F., and Perrot, C. (2017). “Effect of the three-dimensional microstructure on the sound absorption of foams: A parametric study,”

J. Acoust. Soc. Am.142(2), 1130–1140.

Chevillotte, F., Perrot, C., and Guillon, E. (2013). “A direct link between microstructure and acoustical macro-behavior of real double porosity foams,”J. Acoust. Soc. Am.134(6), 4681–4690.

FIG. 14. (Color online) Fiber surrounded by a pore volume.